homework3 - Department of Electrical and Computer...

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1 Department of Electrical and Computer Engineering, Cornell University ECE 407: Physics of Semiconductor and Nanostructures Spring 2009 Homework 3 ` Due on Feb. 10, 2009 at 5:00 PM Suggested Readings: a) Lecture notes b) Chapter 2 in Kittel (Introduction to Solid State Physics) Problem 3.1 (A 2D lattice with a basis) Consider the following 2D lattice made of 2 different kinds of atoms (red-unfilled and black-filled), as shown: a) Find the primitive lattice vectors, 1 a r and 2 a r , and all the basis vectors and draw them on the figure above (and submit this figure with your homework). b) Find the area of the Wigner-Seitz primitive cell of the direct lattice. c) Sketch, on the figure above, the Wigner-Seitz primitive cell and submit it with your homework. How many atoms are there in one primitive cell? d) Find the reciprocal lattice vectors: 1 b r and 2 b r . e) Find the area of the first Brillouin zone in k-space. a 2 3 a a x y
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2 f) Sketch the reciprocal lattice points and the first Brillouin zone in k-space in a neat and clean diagram. Also draw the reciprocal lattice primitive vectors. Identify the reciprocal lattice type from among the 5 possible lattice types in 2D. Problem 3.2 (Brillouin zones and energy bands for a hexagonal lattice in 2D) Consider the following 2D hexagonal lattice from your last homework along with the primitive lattice vectors: The corresponding reciprocal lattice is shown below: a) Draw enough Bragg lines on the reciprocal lattice to correctly identify the first and the second Brillouin zones. You can draw them on the homework copy and submit. b) On the same diagram as in part (a) label all regions that belong to the first BZ and label all regions that
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homework3 - Department of Electrical and Computer...

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